Minimally Distant Sets of Lattice Points

We consider the problem of finding two sets of given cardinalities in certain grid graphs, so as to minimize the cross-distance between them. (This is the maximum Manhattan distance between points, one of the first set and another of the second set.) The question is answered completely for grids that are a product of chains of even edge length, and also for the n-cube, by way of the Clements-Lindstrِm approach.